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QPALM: A Proximal Augmented Lagrangian Method for Nonconvex Quadratic Programs

Ben Hermans, Andreas Themelis, Panagiotis Patrinos

Abstract

We propose QPALM, a nonconvex quadratic programming (QP) solver based on the proximal augmented Lagrangian method. This method solves a sequence of inner subproblems which can be enforced to be strongly convex and which therefore admit a unique solution. The resulting steps are shown to be equivalent to inexact proximal point iterations on the extended-real-valued cost function, which allows for a fairly simple analysis where convergence to a stationary point at an \(R\)-linear rate is shown. The QPALM algorithm solves the subproblems iteratively using semismooth Newton directions and an exact linesearch. The former can be computed efficiently in most iterations by making use of suitable factorization update routines, while the latter requires the zero of a monotone, one-dimensional, piecewise affine function. QPALM is implemented in open-source C code, with tailored linear algebra routines for the factorization in a self-written package LADEL. The resulting implementation is shown to be extremely robust in numerical simulations, solving all of the Maros-Meszaros problems and finding a stationary point for most of the nonconvex QPs in the Cutest test set. Furthermore, it is shown to be competitive against state-of-the-art convex QP solvers in typical QPs arising from application domains such as portfolio optimization and model predictive control. As such, QPALM strikes a unique balance between solving both easy and hard problems efficiently.

QPALM: A Proximal Augmented Lagrangian Method for Nonconvex Quadratic Programs

Abstract

We propose QPALM, a nonconvex quadratic programming (QP) solver based on the proximal augmented Lagrangian method. This method solves a sequence of inner subproblems which can be enforced to be strongly convex and which therefore admit a unique solution. The resulting steps are shown to be equivalent to inexact proximal point iterations on the extended-real-valued cost function, which allows for a fairly simple analysis where convergence to a stationary point at an -linear rate is shown. The QPALM algorithm solves the subproblems iteratively using semismooth Newton directions and an exact linesearch. The former can be computed efficiently in most iterations by making use of suitable factorization update routines, while the latter requires the zero of a monotone, one-dimensional, piecewise affine function. QPALM is implemented in open-source C code, with tailored linear algebra routines for the factorization in a self-written package LADEL. The resulting implementation is shown to be extremely robust in numerical simulations, solving all of the Maros-Meszaros problems and finding a stationary point for most of the nonconvex QPs in the Cutest test set. Furthermore, it is shown to be competitive against state-of-the-art convex QP solvers in typical QPs arising from application domains such as portfolio optimization and model predictive control. As such, QPALM strikes a unique balance between solving both easy and hard problems efficiently.

Paper Structure

This paper contains 38 sections, 79 equations, 8 figures, 4 tables, 7 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Notation
  4. Nonconvex subdifferential and proximal mapping
  5. Paper outline
  6. Proximal ALM
  7. Inexact proximal point
  8. Convergence of Algorithm \ref{['alg:PALM']} for nonconvex QPs
  9. Subproblem minimization
  10. Semismooth Newton method
  11. Exact linesearch
  12. Linear algebra code
  13. Solving linear systems
  14. KKT system
  15. Schur system
  16. ...and 23 more sections

Figures (8)

  • Figure 1: Runtime comparison of KKT and Schur complement methods when applying QPALM to the Maros Meszaros test set.
  • Figure 2: Performance profile for QPALM and IPOPT on the nonconvex QPs of the Cutest test set where both converged to the same approximate stationary point.
  • Figure 3: Performance profiles comparing QPALM with OSQP and Gurobi respectively on the Maros Meszaros problem set for a tolerance of $10^{-3}$.
  • Figure 4: Performance profiles comparing QPALM with OSQP and Gurobi respectively on the Maros Meszaros problem set for a tolerance of $10^{-6}$.
  • Figure 5: Runtimes of QPALM, OSQP, qpOASES and Gurobi when solving portfolio optimization problems of varying sizes. The tolerances are $10^{-3}$ and $10^{-6}$ left and right respectively.
  • ...and 3 more figures

Theorems & Definitions (4)

  • proof
  • proof
  • proof
  • proof